Event-triggered bipartite consensus control algorithm design for nonlinear second-order multi-agent systems

Abstract

Bipartite consensus of nonlinear second-order multi-agent systems with directed communication topology is investigated in this article. A kind of novel event-triggered control protocol is proposed, which based on relative information sensing. Different from some conventional event-triggered control for second-order multi-agent systems, the event-triggered control in distributed way for second-order multi-agent systems is studied in this article, which imply that the controller of each follower only needs to collect the local information. To eliminate too large control gain, the control gain of control protocol is further designed based on adaptive way. Moreover, these control algorithms prove that Zeno behavior can be avoided. Finally, the performance of the proposed control protocols is presented by virtue of two simulation examples.

Keywords

Multi-agent systems event-triggered control relative information sensing adaptive control bipartite consensus

Introduction

Due to the wide application range of cooperative control of multi-agent systems, it has attracted broad attention of many researchers in recent years. The aim of cooperative control for multi-agent systems is to design appropriate control algorithm to make multi-agent systems to fulfill complex tasks, such as formation control (Hu et al., 2020; Liu et al., 2020; Tang et al., 2021), containment control (Li et al., 2021b; Liang et al., 2021; Wang et al., 2020), clusters consensus control (Hou et al. 2015; Xu et al., 2016), and flocking control (Lu et al., 2011; Zhu et al., 2013). In the research of cooperative control, consensus control is one of the fundamental research topics of cooperative control for multi-agent systems, which enables all agents in the multi-agent systems to achieve an agreement on states. In Saber and Murray (2004), according to the knowledge of algebraic graph theory, matrix theory, and control theory, consensus problems in networks of agents with different topologies were studied, and the theoretical framework that multi-agent systems can achieve consensus was obtained. The controllability of linear multi-agent systems under switching topologies was analyzed in Lu et al. (2020), and the theoretical framework for determining controllability is proposed. The observer-based control protocol for second-order multi-agent systems under switching topologies was investigated in Ajwad et al. (2021). The formation control issue was discussed in Liu et al. (2021), and a multi-step control algorithm was proposed for linear multi-agent systems under directed switching topologies. In practical application, the relationship among agents is not only cooperation but also competition. However, in the research works of consensus control for multi-agent systems, competition is usually not taken into account. In Altafini (2013), the antagonistic interactions were first considered networks and then generalized to the research of bipartite consensus for multi-agent systems. For multi-agent systems, the relationship of cooperation and competition can be simultaneously applied to complete more complex works. For instance, the bipartite consensus control for networked robotic systems was investigated in Ding et al. (2021). As an extension of the research on consensus control, the research on bipartite consensus control of multi-agent systems has also attracted much attention in recent years, and many research results have been obtained. In Han and Zheng (2021), state feedback control and output feedback control were both discussed to achieve bipartite output consensus for heterogeneous multi-agent systems. In Lu et al. (2021), two finite-time control protocols were proposed, in which the homogeneous function theory was considered to deal with the bipartite consensus problem of integrator-type multi-agent systems with detail-balanced antagonistic interactions. The bipartite fixed-time output consensus problem of heterogeneous multi-agent systems was considered in Zhang et al. (2021), and the control protocols based on two kinds of bipartite fixed-time observers were proposed.

As a key topic in the field of system control, controller design has always attracted the attention of researchers. In practical applications, it is inevitable that the computation abilities and communication channel bandwidth are limited. To deal with the problems, discontinuous control strategies have received a lot of attention in recent years. As one of the main branches of discontinuous control strategies, the characteristic of event-triggered control strategy, that is, the update of sampling information, is based on the predefined event-triggered conditions, rather than the predefined instants. In Cheng et al. (2021), the observer-based event-triggered output feedback control law was investigated for uncertain nonlinear systems to achieve globally asymptotic stabilization. An event-triggered control method for stochastic nonlinear systems with unmeasured states and unknown backlash-like hysteresis was studied in Zhu et al. (2021). A new event-triggered control method was designed by virtue of the estimated states by the established fuzzy observer. Event-triggered control problem for continuous-time stochastic nonlinear delay systems was investigated in Zhu (2019), and the input-to-state practically exponential mean-square stability of stochastic systems with exogenous disturbances by the application of the proposed event-triggered feedback control method. Event-triggered impulsive controller for stochastic delayed systems was proposed in Peng et al. (2021), which is based on event-triggered mechanism and impulse control gain. Meanwhile, the application of event-triggered control strategy in multi-agent systems has also obtained many research results (Deng et al., 2021; Li et al., 2021a; Qian et al., 2019). For the case of directed graph, the adaptive event-triggered consensus problems were systematically discussed for linear multi-agent systems in Li et al. (2021a). The event-triggered output consensus problems were investigated for heterogeneous linear multi-agent systems in Qian et al. (2019) and Deng et al. (2021).

The observer-based adaptive fuzzy output-feedback control method was studied for strict-feedback nonlinear systems to achieve semiglobally uniformly ultimately boundedness in Tong et al. (2020), which is based on adaptive backstepping recursive design. For fractional-order nonlinear systems, the adaptive control problem was investigated in Li et al. (2021c), and the convergence can be guaranteed using the proposed adaptive controller. Adaptive strategy was studied in Song et al. (2021), the adaptive model predictive controller was designed, and the control horizon can be adaptively adjusted. Adaptive strategy has also been applied in multi-agent systems (Guo and Luo, 2018; Li et al., 2018). In Guo and Luo (2018), a novel adaptive–impulsive protocol was proposed by combining adaptive strategy and impulsive control strategy for multi-agent systems. The edge-based adaptive control protocol was studied in Li et al. (2018), which addressed the consensus problem of multi-agent systems.

Up to now, cooperative control and reduction of resource consumption have become hot topics in the research of multi-agent systems. With this background, the event-triggered control problem of nonlinear second-order multi-agent systems is discussed in this article. The main contributions of this article are summarized as follows:

A class of novel event-triggered control protocols are studied for nonlinear second-order multi-agent systems. Different from many existing results, such as Li et al. (2014), Liu et al. (2016), and Wang et al. (2017), the relative information sensing is used for information sampling of the proposed controllers.

Distributed event-triggered control of linear multi-agent systems is studied in Li et al. (2020). Different from the research results, a new distributed event-triggered control protocol with fixed control gain is proposed for nonlinear second-order multi-agent systems.

A new distributed event-triggered control protocol with adaptive control gain. The system consumption can be reduced due to the application of adaptive way. Thus, the control gains can be adjusted according to the evolution of the system states.

The remainder of this article is structured as follows. Some graph theory notations and problem formulation are presented in section “Preliminaries.” The novel distributed event-triggered bipartite consensus control protocol and adaptive event-triggered bipartite consensus control protocol are proposed in section “Main results.” In section “Numerical simulations,” two simulation examples are provided to verify the validity of theoretical results. Finally, the paper is summarized in section “Conclusion.”

Notation. $R^{n \times m}$ and $R^{n}$ represent $n \times m -$ real matrices and $n -$ vectors, respectively. The $n \times n$ identity matrix is denoted by $I_{n}$ . The maximum and minimum eigenvalues of a square matrix are denoted by $λ_{max} (\cdot)$ , $λ_{min} (\cdot)$ , respectively. $diag {\cdot, \cdot}$ stands for a diagonal matrix. ⊗ is the Kronecker product, and $| | \cdot | |$ is the Euclidean norm for vectors.

Preliminaries

Graph theory notations

The multi-agent systems with $N + 1$ nodes are considered in this paper. The communication topology among the followers can be described as a weighted directed signed graph $G = {V, E, A}$ . $V = {v_{1}, \dots, v_{N}}$ represents the set of nodes, and the nodes are labeled as $1, \dots, N$ . $E \subseteq V \times V$ means the edge set. If information can be transmitted from follower $j$ to follower $i$ , $(v_{j}, v_{i}) \in E$ , otherwise $(v_{j}, v_{i}) \notin E$ . The neighbor set of node $v_{i}$ is defined by $N_{i} = {j \in V | (v_{j}, v_{i}) \in E}$ . The connectivity matrix of the graph is $A = [a_{ij}]_{N \times N}$ , when $(v_{j}, v_{i}) \in E$ , $a_{ij} \neq 0$ , and $a_{ij} = 0$ otherwise. $L = [l_{ij}]_{N \times N}$ denotes the Laplacian matrix of graph $G$ , the values of the elements are $l_{ii} = \sum_{j = 1}^{N} | a_{ij} |$ and $l_{ij} = - a_{ij} (i \neq j)$ . The diagonal matrix $B = diag {b_{1}, \dots, b_{N}}$ means the connection relationship between the leader and the followers, when follower $i$ can directly receive information from the leader, $b_{i} > 0$ and $b_{i} = 0$ otherwise. The graph matrix is defined as $H = L + B$ .

Definition 1. (Wen et al., 2018) For a signed digraph, structurally balanced means that all the nodes are admitted to be divided into two clusters: $V_{1}$ and $V_{2}$ . The two clusters satisfy the conditions that $V_{1} \cup V_{2} = V$ , $V_{1} \cap V_{2} = \emptyset$ , that is, if $v_{i}, v_{j} \in V_{q}$ , $a_{ij} \geq 0$ , if $v_{i} \in V_{q}$ , $v_{j} \in V_{3 - q}$ , $a_{ij} \leq 0 (q \in {1, 2})$ . Otherwise, the signed digraph is structurally unbalanced.

Lemma 1. (Wen et al., 2018) Define a set $D = {Σ = diag {σ_{1}, \dots, σ_{N}}, σ_{i} \in {\pm 1}}, i = 1, \dots, N$ . For a given signed graph $G$ , if there exists a diagonal matrix $Σ \in D$ , all the elements of $\bar{A} = Σ A Σ$ are non-negative, the signed graph $G$ is called structurally balanced. In addition, the two clusters of $G$ are determined by $Σ$ , that is, if $v_{i} \in V_{1}$ , $σ_{i} = 1$ and $v_{i} \in V_{2}$ , $σ_{i} = - 1$ .

Assumption 1. The fixed signed digraph $\bar{G}$ discussed in this paper satisfies the following conditions:

The graph $\bar{G}$ is structurally balanced.

The graph $\bar{G}$ contains a spanning tree, and the leader is regarded as the root node.

For the convenience of the analysis in what follows, the matrix can be defined that $\hat{H} = Σ H Σ$ .

Problem formulation

The leader-following bipartite consensus problem of nonlinear second-order multi-agent systems is discussed in this paper. $N + 1$ agents are considered, which contain a leader and $N$ followers with nonlinear second-order dynamics.

The dynamics of each follower

The $N$ followers are indexed by $1, \dots, N$ , the dynamics of the $i th$ follower are described by

{\begin{matrix} {\dot{x}}_{i} (t) = v_{i} (t), \\ {\dot{v}}_{i} = σ_{i} f (t, σ_{i}^{- 1} x_{i} (t), σ_{i}^{- 1} v_{i} (t)) + u_{i} (t), i = 1, \dots, N \end{matrix}

(1)

where $x_{i}, v_{i} \in R^{n}$ represent the position and velocity state variables of $i th$ follower, respectively. The intrinsic dynamics of $i th$ follower can be described by a continuously nonlinear function $f : R^{+} \times R^{n} \times R^{n} \to R^{n}$ . $σ_{i}$ is the element of matrix $Σ$ , and $Σ = diag {σ_{1}, \dots, σ_{N}}$ .

The dynamics of the leader

The dynamics of the leader can be described by

{\begin{matrix} {\dot{x}}_{d} (t) = v_{d} (t), \\ {\dot{v}}_{d} = f (t, x_{d} (t), v_{d} (t)) \end{matrix}

(2)

where $x_{d}, v_{d} \in R^{n}$ represent the position and velocity state variables, respectively.

The objective of this paper is to ensure the achievement of bipartite consensus for the multi-agent systems (1) and (2). Therefore, the distributed event-triggered control algorithm is discussed. From a mathematical viewpoint, for any initial states, the states of equations (1) and (2) satisfy $lim_{t \to \infty} ‖ x_{i} (t) - σ_{i} x_{d} (t) ‖ = 0$ , $lim_{t \to \infty} ‖ v_{i} (t) - σ_{i} v_{d} (t) ‖ = 0, i = 1, \dots, N$ .

Assumption 2. (Qin et al. 2019) For the vectors $x, x', v, v' \in R^{n}$

‖ f (t, x, v) - f (t, x', v') ‖ \leq ρ_{1} ‖ x - x' ‖ + ρ_{2} ‖ v - v' ‖, \forall t \geq 0

where $ρ_{1}$ and $ρ_{2}$ are non-negative constants.

Lemma 2. (Wang et al., 2016) For any given positive-definite matrix $Q \in R^{n \times n}$ , then

2 x^{T} y \leq x^{T} Qx + y^{T} Q^{- 1} y

where $x, y \in R^{n}$ are any vectors.

Main results

For the $i th$ follower, define the combined measurement errors as

{\tilde{x}}_{i} (t) = \sum_{j \in N_{i}} | a_{ij} | (x_{i} (t) - sign (a_{ij}) x_{j} (t)) + b_{i} (x_{i} (t) - σ_{i} x_{d} (t))

(3a)

{\tilde{v}}_{i} (t) = \sum_{j \in N_{i}} | a_{ij} | (v_{i} (t) - sign (a_{ij}) v_{j} (t)) + b_{i} (v_{i} (t) - σ_{i} v_{d} (t))

(3b)

where equations (3a) and (3b) are the combined measurement errors of position and velocity, respectively.

In this paper, one of the motivations is to investigate how to reduce the communication burden among agents. Therefore, the event-triggered control algorithms are introduced to avoid continuous consumption of communication resources. The sampling instants of event-triggered control algorithm belong to the discrete-time sequence ${t_{k}^{i}}_{k = 0, 1, 2, \dots}, i = 1, \dots, N$ . For $i th$ follower, the sampling instant $t_{k}^{i}$ is determined by event-triggered conditions, and the control input is updated at the instant.

Define the tracking errors of position and velocity for $i th$ follower as ${\bar{x}}_{i} = x_{i} - σ_{i} x_{d}$ , ${\bar{v}}_{i} = v_{i} - σ_{i} v_{d}$ . Therefore, the combined measurement errors can be rewritten as

{\tilde{x}}_{i} = \sum_{j = 1}^{N} l_{ij} {\bar{x}}_{j} + b_{i} {\bar{x}}_{i}

(4a)

{\tilde{v}}_{i} = \sum_{j = 1}^{N} l_{ij} {\bar{v}}_{j} + b_{i} {\bar{v}}_{i}

(4b)

By combining equations (1) and (2), one has

{\begin{matrix} {\overset{\cdot}{\bar{x}}}_{i} = {\bar{v}}_{i}, \\ {\overset{\cdot}{\bar{v}}}_{i} = σ_{i} f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - σ_{i} f (t, x_{d}, v_{d}) + u_{i} \end{matrix}

(5)

By taking error into consideration, vectors of position and velocity are denoted by $\bar{x} = [{\bar{x}}_{1}^{T}, \dots, {\bar{x}}_{N}^{T}]^{T}$ and $\bar{v} = [{\bar{v}}_{1}^{T}, \dots, {\bar{v}}_{N}^{T}]^{T}$ , and the corresponding compact form of equation (5) is given by

{\begin{matrix} \overset{\cdot}{\bar{x}} = \bar{v}, \\ \overset{\cdot}{\bar{v}} = \bar{F} (t, x, v) + u \end{matrix}

(6)

And

$\bar{F} (t, x, v) = (Σ \otimes I_{n}) ([\begin{matrix} f (t, σ_{1}^{- 1} x_{1}, σ_{1}^{- 1} v_{1}) \\ ⋮ \\ f (t, σ_{N}^{- 1} x_{N}, σ_{N}^{- 1} v_{N}) \end{matrix}] - 1_{N} \otimes f (t, x_{d}, v_{d}))$ where $u = [u_{1}^{T}, \dots, u_{N}^{T}]^{T}$ .

Distributed event-triggered bipartite consensus

The proposed distributed event-triggered control protocol and the corresponding triggering condition are introduced in this subsection. The event-triggered control protocols discussed in this paper are based on relative information sensing and distributed way. When $t \geq t_{k}^{i}$ , the next sampling instant $t_{k + 1}^{i}$ is determined when the triggering function $f_{i} < 0$ , that is, $t_{k + 1}^{i} = inf_{t > t_{k}^{i}} {t | f_{i} (e_{x_{i}} (t), e_{v_{i}} (t), {\tilde{x}}_{i} (t), {\tilde{v}}_{i} (t), t) = 0}, k \in N$ where $e_{x_{i}}$ and $e_{v_{i}}$ denote the measurement errors of position and velocity, which are defined as $e_{x_{i}} (t) = {\tilde{x}}_{i} (t_{k_{t}^{i}}^{i}) - {\tilde{x}}_{i} (t), e_{v_{i}} (t) = {\tilde{v}}_{i} (t_{k_{t}^{i}}^{i}) - {\tilde{v}}_{i} (t)$ and the last event at the instant $t$ is expressed as $k_{t}^{i} = \underset{r}{\arg} min_{r \in N; t \geq t_{r}^{i}} {t - t_{r}^{i}}$

Thus, when $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , $t_{k_{t}^{i}}^{i} = t_{k}^{i}$ . The following function is considered for $i th$ follower

\begin{matrix} f_{i} = ({‖ e_{x_{i}} (t) ‖}^{2} + {‖ e_{v_{i}} (t) ‖}^{2}) - α_{i} ({‖ {\tilde{x}}_{i} (t) ‖}^{2} + {‖ {\tilde{v}}_{i} (t) ‖}^{2}) \\ - \frac{μ}{c} \exp (- ζ t) \end{matrix}

(7)

where $μ$ and $ζ$ are known positive constants. Furthermore, the distributed event-triggered control protocol for $i th$ follower is considered as the following form

u_{i} (t) = - c ({\tilde{x}}_{i} (t_{k_{t}^{i}}^{i}) + {\tilde{v}}_{i} (t_{k_{t}^{i}}^{i}))

(8)

When $t \in [t_{k}^{i}, t_{k + 1}^{i}), i = 1, \dots, N, k \in N$ , equation (8) can be further calculated as

\begin{matrix} u_{i} (t) = - c ({\tilde{x}}_{i} (t_{k}^{i}) + {\tilde{v}}_{i} (t_{k}^{i})) \\ = - c ({\tilde{x}}_{i} (t) + {\tilde{v}}_{i} (t)) - c (e_{x_{i}} (t) + e_{v_{i}} (t)) \\ = - c (\sum_{j = 1}^{N} h_{ij} {\bar{x}}_{j} (t) + \sum_{j = 1}^{N} h_{ij} {\bar{v}}_{j} (t)) - c (e_{x_{i}} (t) + e_{v_{i}} (t)) \end{matrix}

(9)

where $h_{ij}$ is the element of matrix $H$ , that is, $H = [h_{ij}]_{N \times N}$ . Define the vectors $e_{x} (t) = [e_{x_{1}}^{T} (t), \dots, e_{x_{N}}^{T} (t)]^{T}$ , $e_{v} (t) = {[e_{v_{1}}^{T} (t), \dots, e_{v_{N}}^{T} (t)]}^{T}$ . Consequently, the corresponding compact form of equation (9) is written as

u = - c ((H \otimes I_{n}) \bar{x} + (H \otimes I_{n}) \bar{v}) - c (e_{x} + e_{v})

(10)

Substituting equation (10) into equation (6), one has

{\begin{matrix} \overset{\cdot}{\bar{x}} = \bar{v}, \\ \overset{\cdot}{\bar{v}} = \bar{F} (t, x, v) - c ((H \otimes I_{n}) \bar{x} + (H \otimes I_{n}) \bar{v}) - c (e_{x} + e_{v}) \end{matrix}

In what followers, the theorem is summarized as follows. The sufficient conditions for the effectiveness of the control protocol (8) under triggering condition (7) are provided in the theorem.

Remark 1. The event-triggered control protocol proposed in this article is based on relative information sensing. Compared with absolute information sensing, selecting relative information sensing can effectively reduce the sampling frequency of the controller. For example, the event-triggered instants for the followers $i$ and $j$ are presented in Figure 1. If absolute information sensing is used in the control protocol, one has

\begin{matrix} {\tilde{x}}_{i} (t_{k}^{i}) = \sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{i} (t_{k}^{i}) - {\bar{x}}_{j} (t_{k'}^{j})) + b_{i} {\bar{x}}_{i} (t_{k}^{i}) \\ {\tilde{v}}_{i} (t_{k}^{i}) = \sum_{j \in N_{i}} a_{ij} ({\bar{v}}_{i} (t_{k}^{i}) - {\bar{v}}_{j} (t_{k'}^{j})) + b_{i} {\bar{v}}_{i} (t_{k}^{i}) \end{matrix}

where $k' = \underset{r}{\arg} min_{r \in N; t \geq t_{r}^{j}} {t - t_{r}^{j}}$ . Obviously, when $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , $t_{k'}^{j}$ has changed three times from $t_{k}^{i}$ to $t_{k + 1}^{i}$ . When relative information sensing is used, the controller for the follower $i$ only samples once.

Remark 2. Up to now, a large number of research results of event-triggered cooperative control were acquired for linear and integrator-type multi-agent systems. In this article, a more complex multi-agent system dynamics is investigated, and bipartite consensus can be achieved using the designed event-triggered control protocol. When nonlinear function $f$ exists in the second-order dynamics, the keys to achieve bipartite consensus depend on the selection of control gain $c$ and parameters $α_{i}$ .

Theorem 1. When Assumptions 1 and 2 hold, consider a class of nonlinear second-order multi-agent systems (1) and (2) under the control law (8). The sufficient conditions to ensure the achievement of leader-following bipartite consensus are proffered as

$λ_{min} ({\hat{H}}^{T} + \hat{H}) > 0$

$\begin{matrix} η_{1} = [\begin{matrix} 2 p I_{N} - c ({\hat{ℋ}}^{T} + \hat{ℋ}) + c (ϖ + \frac{2 N β}{ϖ}) I_{N} \end{matrix} \to \\ * \\ \leftarrow \begin{matrix} (κ - c) ({\hat{ℋ}}^{T} + \hat{ℋ}) + ϖ c I_{N} \\ 2 q I_{N} + 2 I_{N} - c ({\hat{ℋ}}^{T} + \hat{ℋ}) + c (ϖ + \frac{2 N β}{ϖ}) I_{N} \end{matrix}] > 0 \end{matrix}$

$ζ \neq \frac{λ_{max} (η_{1})}{2 λ_{max} (Ω)}$

where $p = 1.5 ρ_{1} + 0.5 ρ_{2}$ , $q = 0.5 ρ_{1} + 1.5 ρ_{2}$ . $κ$ and $ϖ$ are positive constants. Furthermore, the Zeno behavior can be avoided.

Proof. In the first step, the achievement of global bipartite consensus is proved. The candidate Lyapunov function is constructed as

\begin{array}{l} V (t) = \frac{1}{2} [{\bar{x}}^{T} (t), {\bar{v}}^{T} (t)] ([\begin{matrix} κ (ℋ^{T} + ℋ) & I_{N} \\ I_{N} & I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} \bar{x} (t) \\ \bar{v} (t) \end{matrix}] \\ = \frac{1}{2} [{\bar{x}}^{T} (t), {\bar{v}}^{T} (t)] ([\begin{matrix} Σ & 0_{N \times N} \\ 0_{N \times N} & Σ \end{matrix}] \otimes I_{n}) \times \\ \times ([\begin{matrix} κ ({\hat{ℋ}}^{T} + \hat{ℋ}) & I_{N} \\ I_{N} & I_{N} \end{matrix}] \otimes I_{n}) \times \\ \times ([\begin{matrix} Σ & 0_{N \times N} \\ 0_{N \times N} & Σ \end{matrix}] \otimes I_{n}) [\begin{matrix} \bar{x} (t) \\ \bar{v} (t) \end{matrix}] \\ = \frac{1}{2} [{\hat{x}}^{T} (t), {\hat{v}}^{T} (t)] (Ω \otimes I_{n}) [\begin{matrix} \hat{x} (t) \\ \hat{v} (t) \end{matrix}], \end{array}

where $\hat{x} (t) = (Σ \otimes I_{n}) \bar{x} (t)$ , $\hat{v} (t) = (Σ \otimes I_{n}) \bar{v} (t)$ , and $Ω = [\begin{matrix} κ ({\hat{H}}^{T} + \hat{H}) & I_{N} \\ I_{N} & I_{N} \end{matrix}]$

Therefore, defining ${\hat{e}}_{x} = (Σ \otimes I_{n}) e_{x}$ , ${\hat{e}}_{v} = (Σ \otimes I_{n}) e_{v}$ , one has

{\begin{matrix} \overset{\cdot}{\hat{x}} = \hat{v} \\ \overset{\cdot}{\hat{v}} = (Σ \otimes I_{n}) \bar{F} (t, x, v) - c ((\hat{H} \otimes I_{n}) \hat{x} + (\hat{H} \otimes I_{n}) \hat{v}) - c ({\hat{e}}_{x} + {\hat{e}}_{v}) \end{matrix}

Namely

\begin{matrix} [\begin{matrix} \overset{\cdot}{\hat{x}} \\ \overset{\cdot}{\hat{v}} \end{matrix}] = [\begin{matrix} 0_{Nn \times 1} \\ (Σ \otimes I_{n}) \bar{F} (t, x, v) \end{matrix}] \\ - [\begin{matrix} \hat{v} \\ c ((\hat{H} \otimes I_{n}) (\hat{x} + \hat{v}) + ({\hat{e}}_{x} + {\hat{e}}_{v})) \end{matrix}] \\ = [\begin{matrix} 0_{Nn \times 1} \\ (Σ \otimes I_{n}) \bar{F} (t, x, v) \end{matrix}] \\ + ([\begin{matrix} 0_{N \times N} & I_{N} \\ - c \hat{H} & - c \hat{H} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ - ([\begin{matrix} 0_{N \times N} & 0_{N \times N} \\ c I_{N} & c I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}] \end{matrix}

(11)

Therefore, taking the time derivative of $V (t)$ along equation (11) yields

\begin{matrix} \overset{\cdot}{V} = [{\hat{x}}^{T}, {\hat{v}}^{T}] (Ω \otimes I_{n}) [\begin{matrix} \overset{\cdot}{\hat{x}} \\ \overset{\cdot}{\hat{v}} \end{matrix}] \\ = [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} κ \overset{⌣}{H} & I_{N} \\ I_{N} & I_{N} \end{matrix}] \otimes I_{n}) ([\begin{matrix} 0_{Nn \times 1} \\ (Σ \otimes I_{n}) \bar{F} (t, x, v) \end{matrix}] \\ + ([\begin{matrix} 0_{N \times N} & I_{N} \\ - c \hat{H} & - c \hat{H} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ - ([\begin{matrix} 0_{N \times N} & 0_{N \times N} \\ c I_{N} & c I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}]) \\ = [{\hat{x}}^{T}, {\hat{v}}^{T}] [\begin{matrix} (Σ \otimes I_{n}) \bar{F} (t, x, v) \\ (Σ \otimes I_{n}) \bar{F} (t, x, v) \end{matrix}] \\ + [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} - c \hat{H} & κ \overset{⌣}{H} - c \hat{H} \\ - c \hat{H} & I_{N} - c \hat{H} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ + [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} - c I_{N} & - c I_{N} \\ - c I_{N} & - c I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}] \end{matrix}

(12)

where $\overset{⌣}{H} = {\hat{H}}^{T} + \hat{H}$ .

Furthermore, it can be obtained that

\begin{matrix} [{\hat{x}}^{T}, {\hat{v}}^{T}] [\begin{matrix} (Σ \otimes I_{n}) \bar{F} (t, x, v) \\ (Σ \otimes I_{n}) \bar{F} (t, x, v) \end{matrix}] \\ = {\hat{x}}^{T} (Σ \otimes I_{n}) \bar{F} (t, x, v) + {\hat{v}}^{T} (Σ \otimes I_{n}) \bar{F} (t, x, v) \\ = \sum_{i = 1}^{N} {\hat{x}}_{i}^{T} (f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d})) \\ + \sum_{i = 1}^{N} {\hat{v}}_{i}^{T} (f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d})) \\ \leq \sum_{i = 1}^{N} ‖ {\hat{x}}_{i} ‖ ‖ f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d}) ‖ \\ + \sum_{i = 1}^{N} ‖ {\hat{v}}_{i} ‖ ‖ f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d}) ‖ \end{matrix}

(13)

Based on Assumption 2, the following inequality hold

\begin{matrix} ‖ f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d}) ‖ \\ \leq ρ_{1} ‖ σ_{i}^{- 1} x_{i} - x_{d} ‖ + ρ_{2} ‖ σ_{i}^{- 1} v_{i} - v_{d} ‖ \\ = ρ_{1} ‖ x_{i} - σ_{i} x_{d} ‖ + ρ_{2} ‖ v_{i} - σ_{i} v_{d} ‖ \\ = ρ_{1} ‖ {\hat{x}}_{i} ‖ + ρ_{2} ‖ {\hat{v}}_{i} ‖ \end{matrix}

(14)

Substituting equation (14) into equation (13), it is easy to verify that

\begin{matrix} \sum_{i = 1}^{N} ‖ {\hat{x}}_{i} ‖ ‖ f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d}) ‖ \\ + \sum_{i = 1}^{N} ‖ {\hat{v}}_{i} ‖ ‖ f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d}) ‖ \\ \leq \sum_{i = 1}^{N} ‖ {\hat{x}}_{i} ‖ (ρ_{1} ‖ {\hat{x}}_{i} ‖ + ρ_{2} ‖ {\hat{v}}_{i} ‖) + \sum_{i = 1}^{N} ‖ {\hat{v}}_{i} ‖ (ρ_{1} ‖ {\hat{x}}_{i} ‖ + ρ_{2} ‖ {\hat{v}}_{i} ‖) \\ = ρ_{1} \sum_{i = 1}^{N} {‖ {\hat{x}}_{i} ‖}^{2} + ρ_{2} \sum_{i = 1}^{N} {‖ {\hat{v}}_{i} ‖}^{2} + (ρ_{1} + ρ_{2}) \sum_{i = 1}^{N} ‖ {\hat{x}}_{i} ‖ ‖ {\hat{v}}_{i} ‖ \end{matrix}

(15)

In light of Lemma 2, equation (15) can be further calculated as

\begin{matrix} ρ_{1} \sum_{i = 1}^{N} {‖ {\hat{x}}_{i} ‖}^{2} + ρ_{2} \sum_{i = 1}^{N} {‖ {\hat{v}}_{i} ‖}^{2} + (ρ_{1} + ρ_{2}) \sum_{i = 1}^{N} ‖ {\hat{x}}_{i} ‖ ‖ {\hat{v}}_{i} ‖ \\ \leq (\frac{3}{2} ρ_{1} + \frac{1}{2} ρ_{2}) \sum_{i = 1}^{N} {‖ {\hat{x}}_{i} ‖}^{2} + (\frac{1}{2} ρ_{1} + \frac{3}{2} ρ_{2}) \sum_{i = 1}^{N} {‖ {\hat{v}}_{i} ‖}^{2} \\ = [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} p I_{N} & 0_{N \times N} \\ 0_{N \times N} & q I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \end{matrix}

(16)

Meanwhile, according to Lemma 2, one has

\begin{matrix} [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} - c I_{N} & - c I_{N} \\ - c I_{N} & - c I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}] \\ = - c [{\hat{x}}^{T}, {\hat{v}}^{T}] (℘_{N} \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}] \\ = - c {((\sqrt{℘_{N}} \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}])}^{T} ((\sqrt{℘_{N}} \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}]) \\ \leq \frac{c ϖ}{2} [{\hat{x}}^{T}, {\hat{v}}^{T}] (℘_{N} \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ + \frac{c}{2 ϖ} [{\hat{e}}_{x}^{T}, {\hat{e}}_{v}^{T}] (℘_{N} \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}] \end{matrix}

(17)

where $I_{p} = [I_{p}, I_{p}]$ and $℘_{p} = [I_{p}^{T}, I_{p}^{T}]^{T}$ , $p$ is arbitrary positive constant. Combined with the fact that $λ_{max} (℘_{N}) = 2$ , the following conclusion can be easily obtained

\frac{c}{2 ϖ} [{\hat{e}}_{x}^{T}, {\hat{e}}_{v}^{T}] (℘_{N} \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}] \leq \frac{c}{ϖ} ({‖ {\hat{e}}_{x} ‖}^{2} + {‖ {\hat{e}}_{v} ‖}^{2})

(18)

Note that $H = [H_{1}^{T}, \dots, H_{N}^{T}]^{T}$ , it can further yield

\begin{matrix} {‖ {\tilde{x}}_{i} ‖}^{2} = {‖ ([h_{i 1}, \dots, h_{iN}] \otimes I_{n}) [\begin{matrix} {\bar{x}}_{1} \\ ⋮ \\ {\bar{x}}_{N} \end{matrix}] ‖}^{2} \\ = {\bar{x}}^{T} (H_{i}^{T} H_{i} \otimes I_{n}) \bar{x} \end{matrix}

Similarly, one has $‖ {\tilde{v}}_{i} ‖^{2} = {\bar{v}}^{T} (H_{i}^{T} H_{i} \otimes I_{n}) \bar{v}$

Through the analysis of triggering condition (7), it can be verified that

\begin{matrix} {‖ {\hat{e}}_{x_{i}} ‖}^{2} + {‖ {\hat{e}}_{v_{i}} ‖}^{2} \leq α_{i} ({‖ {\tilde{x}}_{i} ‖}^{2} + {‖ {\tilde{v}}_{i} ‖}^{2}) + \frac{μ}{c} \exp (- ζ t) \\ = α_{i} ({\bar{x}}^{T} (H_{i}^{T} H_{i} \otimes I_{n}) \bar{x} + {\bar{v}}^{T} (H_{i}^{T} H_{i} \otimes I_{n}) \bar{v}) \\ + \frac{μ}{c} \exp (- ζ t) \\ = β ({‖ \bar{x} ‖}^{2} + {‖ \bar{v} ‖}^{2}) + \frac{μ}{c} \exp (- ζ t) \end{matrix}

where $β = α_{i} λ_{max} (H_{i}^{T} H_{i})$ and further implies

‖ {\hat{e}}_{x} ‖^{2} + ‖ {\hat{e}}_{v} ‖^{2} \leq N β ({‖ \bar{x} ‖}^{2} + {‖ \bar{v} ‖}^{2}) + \frac{N μ}{c} \exp (- ζ t)

(19)

Therefore, substituting equations (18) and (19) into equation (17), one has

\begin{matrix} [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} - c I_{N} & - c I_{N} \\ - c I_{N} & - c I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} {\hat{e}}_{x} \\ {\hat{e}}_{v} \end{matrix}] \\ \leq [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} c (\frac{ϖ}{2} + \frac{N β}{ϖ}) I_{N} & \frac{c ϖ}{2} I_{N} \\ * & c (\frac{ϖ}{2} + \frac{N β}{ϖ}) I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ + \frac{N μ}{ϖ} \exp (- ζ t) \end{matrix}

(20)

Thus, it can yield

\begin{matrix} \overset{\cdot}{V} \leq \frac{1}{2} [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} - c \overset{⌣}{H} & (κ - c) \overset{⌣}{H} \\ * & 2 I_{N} - c \overset{⌣}{H} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ + [{\hat{x}}^{T}, {\hat{v}}^{T}] ((\frac{c ϖ}{2} ℘_{N} + \frac{N β}{ϖ} I_{2 N}) \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ + \frac{N μ}{ϖ} \exp (- ζ t) \\ + [{\hat{x}}^{T}, {\hat{v}}^{T}] ([\begin{matrix} p I_{N} & 0_{N \times N} \\ 0_{N \times N} & q I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ = - \frac{1}{2} [{\hat{x}}^{T}, {\hat{v}}^{T}] (η_{1} \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] + \frac{N μ}{ϖ} \exp (- ζ t) \\ \leq - \frac{λ_{min} (η_{1})}{2 λ_{max} (Ω)} V + \frac{N μ}{ϖ} \exp (- ζ t) \end{matrix}

(21)

After calculation, one has

V \leq - \frac{N μ}{ϖ (ζ - \frac{λ_{min} (η_{1})}{2 λ_{max} (Ω)})} (\exp (- ζ t) - \exp (- \frac{λ_{min} (η_{1})}{2 λ_{max} (Ω)} t)) .

Namely

\begin{matrix} lim_{t \to \infty} \int_{0}^{t} V (τ) d_{τ} \leq lim_{t \to \infty} \int_{0}^{t} (- \frac{N μ ν}{ϖ (ζ - \frac{λ_{min} (η_{1})}{2 λ_{max} (\hat{Ω})})}) d_{τ} \\ = \frac{2 λ_{max} (\hat{Ω}) N μ}{ϖ λ_{min} (η_{1}) ζ} < \infty \end{matrix}

(22)

where $ν = (\exp (- ζ τ) - \exp (- \frac{λ_{min} (η_{1})}{2 λ_{max} (\hat{Ω})} τ))$ .

The conclusion that $lim_{t \to \infty} (| | \hat{x} | |^{2} + | | \hat{v} | |^{2}) = 0$ can be obtained from equation (22). Therefore, one has $lim_{t \to \infty} ‖ x_{i} - σ_{i} x_{d} ‖ = 0; lim_{t \to \infty} ‖ v_{i} - σ_{i} v_{d} ‖ = 0$ which implies that the global bipartite leader-following consensus can be achieved. In the next step, it will be proved that Zeno behavior can be avoided under the control protocol (9). Note that

\begin{array}{l} D^{+} ({‖ e_{x_{i}} ‖}^{2} + {‖ e_{v_{i}} ‖}^{2}) \\ = 2 e_{x_{i}}^{T} {\dot{e}}_{x_{i}} + 2 e_{v_{i}}^{T} {\dot{e}}_{v_{i}} \leq 2 ‖ e_{x_{i}} ‖ ‖ {\dot{e}}_{x_{i}} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ {\dot{e}}_{v_{i}} ‖ \\ = 2 ‖ e_{x_{i}} ‖ ‖ {\dot{\tilde{x}}}_{i} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ {\dot{\tilde{v}}}_{i} ‖ \\ = 2 ‖ e_{x_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} {\dot{\bar{x}}}_{j} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} {\dot{\bar{v}}}_{j} ‖ \\ = 2 ‖ e_{x_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} {\bar{v}}_{j} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} ({\bar{f}}_{j} + u_{j}) ‖ \\ = 2 ‖ e_{x_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} {\bar{v}}_{j} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} ({\bar{f}}_{j} \\ - c ({\tilde{x}}_{j} (t_{k}^{j}) + {\tilde{v}}_{j} (t_{k}^{j}))) ‖ \\ \leq 2 ‖ e_{x_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} {\bar{v}}_{j} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{i j} ({\bar{f}}_{j}) ‖ \\ + 2 ‖ e_{v_{i}} ‖ ‖ c \sum_{j = 1}^{N} h_{i j} (({\tilde{x}}_{j} (t_{k}^{j}) + {\tilde{v}}_{j} (t_{k}^{j}))) ‖ \\ = 2 ‖ e_{x_{i}} ‖ ‖ (ℋ \otimes I_{n}) \bar{v} ‖ \\ + 2 ‖ e_{v_{i}} ‖ v ‖ (ℋ \otimes I_{n}) \bar{F} (t, x, v) ‖ + 2 ‖ e_{v_{i}} ‖ Φ_{k} \\ \leq 2 ‖ e_{x_{i}} ‖ ‖ ℋ ‖ ‖ \bar{v} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ ℋ ‖ ‖ \bar{F} (t, x, v) ‖ + 2 ‖ e_{v_{i}} ‖ Φ_{k} \end{array}

(23)

where ${\bar{f}}_{j} = f (t, σ_{j}^{- 1} x_{j}, σ_{j}^{- 1} v_{j}) - f (t, x_{d}, v_{d})$ .

It can be verified that

\begin{matrix} ‖ \bar{F} (t, x, v) ‖ = {(\sum_{i = 1}^{N} {‖ f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d}) ‖}^{2})}^{\frac{1}{2}} \\ \leq \sum_{i = 1}^{N} ‖ f (t, σ_{i}^{- 1} x_{i}, σ_{i}^{- 1} v_{i}) - f (t, x_{d}, v_{d}) ‖ \\ \leq \sum_{i = 1}^{N} (ρ_{1} ‖ {\bar{x}}_{i} ‖ - ρ_{2} ‖ {\bar{v}}_{i} ‖) \\ \leq \sqrt{N} ρ_{1} ‖ \bar{x} ‖ + \sqrt{N} ρ_{2} ‖ \bar{v} ‖ \\ = γ_{1} ‖ \bar{x} ‖ + γ_{2} ‖ \bar{v} ‖ \end{matrix}

(24)

where $γ_{1} = \sqrt{N} ρ_{1}$ and $γ_{2} = \sqrt{N} ρ_{2}$ . Substituting equation (24) into equation (23), one has

\begin{matrix} D^{+} ({‖ e_{x_{i}} ‖}^{2} + {‖ e_{v_{i}} ‖}^{2}) \\ \leq 2 ‖ H ‖ (γ_{1} ‖ e_{v_{i}} ‖ ‖ \bar{x} ‖ + (‖ e_{x_{i}} ‖ + γ_{2} ‖ e_{v_{i}} ‖) ‖ \bar{v} ‖) \\ + 2 ‖ e_{v_{i}} ‖ Φ_{k} \\ \leq 2 ‖ H ‖ (γ_{1} ‖ e_{v} ‖ ‖ \bar{x} ‖ + (‖ e_{x} ‖ + γ_{2} ‖ e_{v} ‖) ‖ \bar{v} ‖) \\ + 2 ‖ e_{v} ‖ Φ_{k} \end{matrix}

(25)

Based on the previous analysis, the conclusion that $lim_{t \to \infty} \int_{0}^{t} V (τ) d_{τ} < \infty$ can be obtained. In other words, the selected Lyapunov function $V (t)$ is bounded. Thus

\begin{matrix} V = \frac{1}{2} [{\bar{x}}^{T}, {\bar{v}}^{T}] ([\begin{matrix} κ (H^{T} + H) & I_{N} \\ I_{N} & I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} \bar{x} \\ \bar{v} \end{matrix}] \\ \geq \frac{1}{2} λ_{min} (Ω) ({‖ \bar{x} ‖}^{2} + {‖ \bar{v} ‖}^{2}), \end{matrix}

which implies that $| | \bar{x} | |$ and $| | \bar{v} | |$ are bounded. Combining equations (4a) and (4b), $\tilde{x}$ , $\tilde{v}$ , $e_{x}$ , and $e_{v}$ are bounded and can be obtained. Therefore, for follower $i$ , the constant $ϵ_{i}$ must be found that satisfies the following inequalities

\begin{matrix} D^{+} ({‖ e_{x_{i}} ‖}^{2} + {‖ e_{v_{i}} ‖}^{2}) \\ \leq 2 ‖ H ‖ (γ_{1} ‖ e_{v} ‖ ‖ \bar{x} ‖ + (‖ e_{x} ‖ + γ_{2} ‖ e_{v} ‖) ‖ \bar{v} ‖) + 2 ‖ e_{v} ‖ Φ_{k} \\ \leq ϵ_{i} < \infty . \end{matrix}

When $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , one has

\begin{matrix} {‖ e_{x_{i}} (t) ‖}^{2} + {‖ e_{v_{i}} (t) ‖}^{2} = \int_{t_{k}^{i}}^{t} D^{+} ({‖ e_{x_{i}} (τ) ‖}^{2} + {‖ e_{v_{i}} (τ) ‖}^{2}) d_{τ} \\ \leq ϵ_{i} (t - t_{k}^{i}) \end{matrix}

(26)

On the one hand, the following inequality can be obtained directly from equation (26)

‖ e_{x_{i}} (t_{k + 1}^{i}) ‖^{2} + ‖ e_{v_{i}} (t_{k + 1}^{i}) ‖^{2} \leq ϵ_{i} (t_{k + 1}^{i} - t_{k}^{i})

(27)

On the other hand, according to equation (7), at trigger instant $t_{k + 1}^{i}$ , one has

\begin{matrix} {‖ e_{x_{i}} (t_{k + 1}^{i}) ‖}^{2} + {‖ e_{v_{i}} (t_{k + 1}^{i}) ‖}^{2} \\ = α_{i} ({‖ {\tilde{x}}_{i} (t_{k + 1}^{i}) ‖}^{2} + {‖ {\tilde{v}}_{i} (t_{k + 1}^{i}) ‖}^{2}) + \frac{μ}{c} \exp (- ζ t_{k + 1}^{i}) \\ \geq \frac{μ}{c} \exp (- ζ t_{k + 1}^{i}) > 0 \end{matrix}

(28)

In light of equations (27) and (28), one has

t_{k + 1}^{i} - t_{k}^{i} \geq \frac{μ}{c ϵ_{i}} \exp (- ζ t_{k + 1}^{i}) > 0

Therefore, the Zeno behavior can be avoided.

Remark 3. The cooperative control of second-order multi-agent systems were investigated in the works of Mu and Wu (2018), Zou et al. (2019), and Liu et al. (2021). In the construction of controller, different control gains are selected for position combinational measurement and velocity combinational measurement, which undoubtedly increases the difficulty of design. Different from these results, the same control gain is used for position combinational measurement and velocity combinational measurement in this article. Therefore, the proposed control algorithm has a simpler design procedure.

Remark 4. Zeno behavior means the phenomenon of infinite number of triggering in a finite time so that the system information is continuously collected. The design principle of event-triggered control algorithm is to make the system information at discrete triggering instants through appropriate triggering conditions. The information continuous sampling is avoided, and thus the resource consumption of the system can be reduced. The avoidance of Zeno behavior must be considered in the design of event-triggered control algorithm.

Adaptive event-triggered bipartite consensus

In this section, it is not only avoiding the continuous consumption of system resources but also the method to deal with too large control gains are investigated. Therefore, the distributed event-triggered control protocol is designed in previous section is extended to adaptive event-triggered control protocol. Specifically, the control protocol for each follower is designed as below

u_{i} (t) = - C (t) ({\tilde{x}}_{i} (t_{k_{t}^{i}}^{i}) + {\tilde{v}}_{i} (t_{k_{t}^{i}}^{i}))

(29)

where $C (t)$ is the adaptive control gain, and the adaptive law of $C (t)$ is given by

\begin{matrix} \overset{\cdot}{C} (t) = \frac{1}{2} θ [\begin{matrix} {\hat{x}}^{T} (t) & {\hat{v}}^{T} (t) \end{matrix}] \times \\ \times ([\begin{matrix} \overset{⌣}{H} & \overset{⌣}{H} \\ * & \overset{⌣}{H} \end{matrix}] \otimes I_{n} - C φ \otimes I_{n}) [\begin{matrix} \hat{x} (t) \\ \hat{v} (t) \end{matrix}] \exp (\partial t) \end{matrix}

(30)

where $θ > 0$ is a known constant, and $φ = \frac{1}{C (0) - \bar{C}} [\begin{matrix} (2 ϖ + \frac{2 N β}{ϖ}) I_{N} & 0_{N \times N} \\ * & (2 ϖ + \frac{2 N β}{ϖ}) I_{N} \end{matrix}]$

According to equations (1) and (2), and control protocol (29), one has

\begin{matrix} \overset{\cdot}{\hat{x}} = \hat{v} \\ \overset{\cdot}{\hat{v}} = (D \otimes I_{n}) \bar{F} (t, x, v) - C ((\hat{H} \otimes I_{n}) (\hat{x} + \hat{v}) + ({\hat{e}}_{x} + {\hat{e}}_{v})) \end{matrix}

(31)

When $t \geq t_{k}^{i}$ , the next sampling instant $t_{k + 1}^{i}$ is determined by the following condition $t_{k + 1}^{i} = \inf_{t > t_{k}^{i}} {t | \overset{⌣}{f}_{i} (e_{x_{i}} (t), e_{v_{i}} (t), {\tilde{x}}_{i} (t), {\tilde{v}}_{i} (t), C (t), t) = 0}, k \in ℕ$

The triggering function ${\overset{⌣}{f}}_{i}$ is designed as follows

\begin{matrix} {\overset{⌣}{f}}_{i} = ({‖ e_{x_{i}} (t) ‖}^{2} + {‖ e_{v_{i}} (t) ‖}^{2}) \\ - α_{i} ({‖ {\tilde{x}}_{i} (t) ‖}^{2} + {‖ {\tilde{v}}_{i} (t) ‖}^{2}) - \frac{μ}{C (t)} \exp (- ζ t) \end{matrix}

(32)

Theorem 2. Under Assumptions 1 and 2, the evolution of the system states of the second-order multi-agent systems (1) and (2) with the adaptive event-triggered control protocol (29) is investigated. If global leader-following bipartite consensus can be achieved, the following conditions must be satisfied.

$λ_{min} ({\hat{H}}^{T} + \hat{H}) > 0$

$\begin{matrix} \begin{matrix} η_{2} = - [\begin{matrix} 2 p I_{N} - \bar{C} ({\hat{H}}^{T} + \hat{H}) \end{matrix} \to \\ * \end{matrix} \\ \leftarrow \begin{matrix} 0_{N \times N} \\ 2 q I_{N} + 2 I_{N} - \bar{C} ({\hat{H}}^{T} + \hat{H}) \end{matrix}] < 0 \end{matrix}$

$ζ \neq ϑ$ .

where $\bar{C}$ , $κ$ , and $ϑ$ are positive constants. Moreover, systems (1) and (2) do not exhibit Zeno behavior.

Proof. First, the candidate Lyapunov function (33) is constructed, which consists of $V_{1} (t)$ and $V_{2} (t)$

V (t) = V_{1} (t) + V_{2} (t)

(33)

and

\begin{matrix} V_{1} (t) = \frac{1}{2} [\begin{matrix} {\bar{x}}^{T} (t) & {\bar{v}}^{T} (t) \end{matrix}] ([\begin{matrix} κ (H^{T} + H) & I_{N} \\ * & I_{N} \end{matrix}] \otimes I_{n}) [\begin{matrix} \bar{x} (t) \\ \bar{v} (t) \end{matrix}] \\ = \frac{1}{2} [\begin{matrix} {\hat{x}}^{T} (t) & {\hat{v}}^{T} (t) \end{matrix}] (Ω \otimes I_{n}) [\begin{matrix} \hat{x} (t) \\ \hat{v} (t) \end{matrix}], \end{matrix}

V_{2} (t) = \frac{{(C (t) - \bar{C})}^{2}}{2 θ} \exp (- \partial t)

Therefore, taking the time derivative of $V_{1} (t)$ along equation (31), one has

{\overset{\cdot}{V}}_{1} \leq \frac{1}{2} [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] ([\begin{matrix} Ξ_{1} & Ξ_{2} \\ * & Ξ_{3} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] + \frac{N μ}{ϖ} \exp (- ζ t)

(34)

where

\begin{matrix} Ξ_{1} = 2 p I_{N} - C ({\hat{H}}^{T} + \hat{H}) + C (ϖ + \frac{2 N β}{ϖ}) I_{N} \\ Ξ_{2} = (κ - C) ({\hat{H}}^{T} + \hat{H}) + C ϖ I_{N} \\ Ξ_{3} = 2 (q + 1) I_{N} - C ({\hat{H}}^{T} + \hat{H}) + C (ϖ + \frac{2 N β}{ϖ}) I_{N} \end{matrix}

Differentiating $V_{2} (t)$ along equation (30), one obtains

\begin{matrix} {\overset{\cdot}{V}}_{2} = \frac{- \partial e^{- \partial t} {(C - \bar{C})}^{2} + 2 e^{- \partial t} (C - \bar{C}) \overset{\cdot}{C}}{2 θ} \\ = - \frac{\partial {(C - \bar{C})}^{2}}{2 θ} \exp (- \partial t) + \frac{1}{2} (C - \bar{C}) [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] \times \\ \times ([\begin{matrix} \overset{⌣}{H} & \overset{⌣}{H} \\ * & \overset{⌣}{H} \end{matrix}] \otimes I_{n} - C φ \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ \leq - \frac{\partial {(C - \bar{C})}^{2}}{2 θ} \exp (- \partial t) \\ + \frac{1}{2} (C - \bar{C}) [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] ([\begin{matrix} \overset{⌣}{H} & \overset{⌣}{H} \\ * & \overset{⌣}{H} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ - \frac{1}{2} (C (0) - \bar{C}) [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] (C φ \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ = - \frac{\partial {(C - \bar{C})}^{2}}{2 θ} \exp (- \partial t) \\ + \frac{1}{2} (C - \bar{C}) [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] ([\begin{matrix} \overset{⌣}{H} & \overset{⌣}{H} \\ * & \overset{⌣}{H} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ - C (ϖ + \frac{N β}{ϖ}) ({‖ \hat{x} ‖}^{2} + {‖ \hat{v} ‖}^{2}) \end{matrix}

(35)

When $κ = \bar{C}$ , combining equations (34) and (35), one obtains

\begin{matrix} \overset{\cdot}{V} = {\overset{\cdot}{V}}_{1} + {\overset{\cdot}{V}}_{2} \\ \leq \frac{1}{2} {\hat{x}}^{T} ((2 p I_{N} - \bar{C} \overset{⌣}{H}) \otimes I_{n}) \hat{x} \\ + \frac{1}{2} {\hat{v}}^{T} ((2 (q + 1) I_{N} - \bar{C} \overset{⌣}{H}) \otimes I_{n}) \hat{v} \\ - \frac{\partial {(C - \bar{C})}^{2}}{2 θ} \exp (- \partial t) + \frac{N μ}{ϖ} \exp (- ζ t) \\ = - \frac{1}{2} [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] (η_{2} \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ - \frac{\partial {(C - \bar{C})}^{2}}{2 θ} \exp (- \partial t) + \frac{N μ}{ϖ} \exp (- ζ t) \\ \leq - \frac{λ_{max} (η_{2})}{2 λ_{max} (Ω)} [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] (Ω \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}] \\ - \frac{\partial {(C - \bar{C})}^{2}}{2 θ} \exp (- \partial t) + \frac{N μ}{ϖ} \exp (- ζ t) \\ \leq - ϑ V + \frac{N μ}{ϖ} \exp (- ζ t) \end{matrix}

(36)

where $ϑ = min {\frac{λ_{max} (η_{2})}{λ_{max} (Ω)}, \partial}$ . The following inequality can be obtained easily,

V \leq - \frac{N μ}{ϖ (ζ - ϑ)} (\exp (- ζ t) - \exp (- ϑ t))

Referring to the calculation method of equation (22), the conclusions $lim_{t \to \infty} | | x_{i} (t) - σ_{i} x_{d} (t) | | = 0$ and $lim_{t \to \infty} | | v_{i} (t) - σ_{i} v_{d} (t) | | = 0$ can be directly obtained in such a case.

Next, we prove that Zeno behavior is avoided. When $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , one has

\begin{matrix} D^{+} ({‖ e_{x_{i}} ‖}^{2} + {‖ e_{v_{i}} ‖}^{2}) \\ \leq 2 ‖ e_{x_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{ij} {\bar{v}}_{j} ‖ \\ + 2 ‖ e_{v_{i}} ‖ ‖ \sum_{j = 1}^{N} h_{ij} (f (t, d_{j}^{- 1} x_{j}, d_{j}^{- 1} v_{j}) - f (t, x_{0}, v_{0})) ‖ \\ + 2 ‖ e_{v_{i}} ‖ ‖ C \sum_{j = 1}^{N} h_{ij} ({\tilde{x}}_{j} (t_{k}^{j}) + {\tilde{v}}_{j} (t_{k}^{j})) ‖ \end{matrix}

(37)

Based on the above analysis, one may directly obtain the conclusions that ${\tilde{x}}_{j} (t_{k}^{j})$ and ${\tilde{v}}_{j} (t_{k}^{j})$ are bounded, and $C (t)$ converges to a fixed value $C_{\max}$ . Therefore, there must be a constant $Φ'_{k}$ and it satisfies

Φ'_{k} \geq ‖ C \sum_{j = 1}^{N} h_{ij} ({\tilde{x}}_{j} (t_{k}^{j}) + {\tilde{v}}_{j} (t_{k}^{j})) ‖

(38)

Substituting equation (38) into equation (37), one has

\begin{matrix} D^{+} ({‖ e_{x_{i}} ‖}^{2} + {‖ e_{v_{i}} ‖}^{2}) \\ \leq 2 ‖ e_{x_{i}} ‖ ‖ H ‖ ‖ \bar{v} ‖ + 2 ‖ e_{v_{i}} ‖ ‖ H ‖ ‖ \bar{F} (t, x, v) ‖ + 2 ‖ e_{v_{i}} ‖ {Φ'}_{k} \end{matrix}

(39)

In light of the calculation in Theorem 1, the following conclusion can be obtained

t_{k + 1}^{i} - t_{k}^{i} \geq \frac{μ}{C (0) ϵ_{i}} \exp (- ζ t_{k + 1}^{i}) > 0, t \in [t_{k}^{i}, t_{k + 1}^{i})

where $ε_{i}$ is a constant with finite value. Obviously, Zeno behavior is avoided.

Remark 5. The distributed event-triggered bipartite consensus problem with fixed gain for second-order multi-agent systems is investigated in the subsection. Because the control algorithm based on fixed gain makes the control gain unable to adjust its value according to the evolution of system states, resulting in unnecessary resource consumption. Therefore, it is necessary extension that the control gain is designed in an adaptive way. The value of the control gain can be adjusted adaptively in the light of the evolution of the system states, so that the control gain will not be too large and will be stable at a certain value after the bipartite consensus is achieved.

Remark 6. According to equation (34), it can be obtained that the matrices $Ξ_{1}$ , $Ξ_{2}$ , and $Ξ_{3}$ all contain the control gain $C (t)$ . Because $C (t)$ is a variable, it will increase the difficulty of the analysis for system state evolution. Therefore, the function $V_{2} (t)$ is added in the Lyapunov function $V (t)$ to eliminate the effect of $C (t)$ . It can be obtained that the term $\frac{1}{2} (C - \bar{C}) [\begin{matrix} {\hat{x}}^{T} & {\hat{v}}^{T} \end{matrix}] ([\begin{matrix} \overset{⌣}{H} & \overset{⌣}{H} \\ * & \overset{⌣}{H} \end{matrix}] \otimes I_{n}) [\begin{matrix} \hat{x} \\ \hat{v} \end{matrix}]$ in equation (35) can eliminate the effect of $C (t)$ in equation (36).

Numerical simulations

The effectiveness of Theorems 1 and 2 is verified by two numerical simulations in this section. The communication topology contains a spanning tree. The adjacency matrix of followers $A$ and the connection relation matrix between the leader and the followers $B$ are presented as

\begin{matrix} A = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ - 1 & 0 & 0 & 1 \\ 0 & - 1 & 0 & 0 \end{matrix}], B = diag {50, 50, 50, 50}, \end{matrix}

where the followers are divided into two clusters: $V_{1} = {1, 2}$ and $V_{2} = {3, 4}$ .

The dynamics considered in this section are equations (1) and (2). Chua’s circuit (Yu et al., 2010) is selected as a nonlinear function for each agent, and the form is as follows

g (t, x (t), v (t)) = [\begin{matrix} ϵ (- v_{1} + v_{2} - ι (v_{1})) \\ v_{1} - v_{2} + v_{3} \\ - ϱ v_{2} \end{matrix}]

where $ι (v_{1}) = ζ_{2} v_{1} + 0.5 (ζ_{1} - ζ_{2}) (| v_{1} + 1 | - | v_{1} - 1 |)$ and $ϵ = 10$ , $ϱ = 18$ , $ζ_{1} = - (\frac{4}{3})$ , $ζ_{2} = - (\frac{3}{4})$ . Because $ρ_{1} = 4.3871$ , $ρ_{2} = 0$ , it is easy to know $p = 6.5807$ , $q = 2.1936$ .

The initial states of each follower are set as: $x_{ij} (t_{0}) \in (- 30, 30)$ , $v_{ij} (t_{0}) \in (- 30, 30)$ , and the initial states of leader are set as: $x_{0 j} (t_{0}) \in (- 3, 3)$ , $v_{0 j} (t_{0}) \in (- 3, 3)$ .

Example 1

Choosing that $κ = 1$ , $ϑ_{1} > 0$ , ensures $λ_{min} (κ ({\bar{H}}^{T} + \bar{H})) > 0$ . After calculation, one has $λ_{min} (ϑ_{1}) = 70.18$ , $ς = 0.05$ .

The state evolution of all agents is illustrated in Figures 2 –4 and the relative position errors between the followers and leader in Figures 5 –7. Obviously, bipartite consensus can be achieved of the second-order multi-agent systems. The triggering instants of each follower are described in Figure 8, defining $h_{i} = | | e_{x_{i}} (t) | |^{2} + | | e_{v_{i}} (t) | |^{2}$ , $g_{i} = α_{i} (| | {\tilde{x}}_{i} (t) | |^{2} + | | {\tilde{v}}_{i} (t) | |^{2}) - (μ / c) \exp (- ζ t)$ , and the instants when the two curves intersect are the triggering instants of each follower.

Example 2

Choosing $κ = 90$ , $\bar{C} = 90$ , $ϖ = 0.5$ , $β = 1$ , and $θ = 1.0 \times 10^{- 3}$ . To ensure $ζ \neq α$ , when $\partial = 0.002$ , the value of $ζ$ is $0.05$ .

The state evolution of all agents is illustrated in Figures 9 –11 and the relative position errors between the followers and leader in Figures 12 –14. Obviously, bipartite consensus can be achieved of the second-order multi-agent systems. The triggering instants of each follower are described in Figure 15, defining $h_{i} = ‖ e_{x_{i}} (t) ‖^{2} + ‖ e_{v_{i}} (t) ‖^{2}$ , $g_{i} = α_{i} ({‖ {\tilde{x}}_{i} (t) ‖}^{2} + {‖ {\tilde{v}}_{i} (t) ‖}^{2}) - \frac{μ}{C (t)} \exp (- ζ t)$ , and the instants when the two curves intersect are the triggering instants of each follower. The trajectory of control gain $C (t)$ is presented in Figure 16. The curve clearly illustrates that when bipartite consensus of multi-agent systems (1) and (2) is achieved, $C (t) \equiv C_{\max}$ , and $C_{\max}$ is a fixed constant.

Conclusion and future work

In this article, the cooperative control problem for second-order multi-agent systems with directed graph are discussed. A class of event-triggered control protocols based on relative information sensing are proposed to achieve bipartite consensus. First, a novel distributed event-triggered control protocol with fixed gain is studied. For the nonlinear second-order multi-agent systems, the bipartite consensus is guaranteed and the resource consumption is effectively reduced. Because the control gain may be too large, the event-triggered control protocol based on adaptive control gain is further studied. Moreover, it has been proved that Zeno behavior does not happen. Finally, the validity of theoretical results is illustrated by two examples.

For nonlinear second-order multi-agent systems, the design of event-triggered control algorithm based on fully distributed way and cooperative control under switching topologies will be research topics in future.

Footnotes

Acknowledgements

The authors thank the anonymous reviewers for their careful reading and constructive comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ruitian Yang

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